2 StructureTheory of Multicriteria Scheduling,Optimality definition,How to solve a multicriteria scheduling problem,Application to a bicriteria scheduling problem,Considerations about the enumeration of optimal solutions.Some models and algorithms,Scheduling with intefering job sets,Scheduling with rejection cost.Solution of bicriteria single machine problem by mathematical programmingVincent T’kindt

3 What is Multicriteria Scheduling?Multicriteria Optimization: How to optimize several conflicting criteria?Scheduling: How to determine the « optimal » allocation of tasks (jobs) to resources (machines) over time?Multicriteria Optimization: How to optimize several conflicting criteria?Scheduling: How to determine the « optimal » allocation of tasks (jobs) to resources (machines) over time?Multicriteria Scheduling =Scheduling + Multicriteria Optimization.Vincent T’kindt

4 Theory of Multicriteria SchedulingWhat about multicriteria optimization?K criteria Zi to minimize,The notion of optimality is definedby means of Pareto optimality,We distinguish between:Strict Pareto optimality,Weak Pareto optimality.Z2Z1A solution x is a strict Pareto optimum iff there does not exist another solution y such that Zi(y) ≤Zi(x), i=1,…,K, with at least one strict inequality.A solution x is a weak Pareto optimum iff there does not exist another solution y such that Zi(y) < Zi(x), i=1,…,K.EWEVincent T’kindt

9 Theory of Multicriteria SchedulingIllustration on an example problem: 1|di|Lmax, CA single machine is available,n jobs have to be processed,pi : processing time,di : due date,Minimize Lmax=maxi(Ci-di) and C=Si Ci,p1C1C21C323123Machinetimed1d2d3Vincent T’kindt

10 Theory of Multicriteria SchedulingIllustration on an example problem: 1|di|Lmax, CDesign of an a posteriori algorithm1A strict Pareto optimum is calculated by means of the e-contraint approachKnown results :The 1||C problem is solved to optimality by Shortest Processing Times first rule (SPT),The 1|di|Lmax problem is solved to optimality by Earliest Due Date first rule (EDD),1 L. van Wassenhove and L.F. Gelders (1980). Solving a bicriterion scheduling problem, EJOR, 4:42-48.Vincent T’kindt

14 Theory of Multicriteria SchedulingCandidate list based algorithm,This a posteriori algorithm is optimal,The scheduling module works in O(nlog(n)),There are at most n(n+1)/2 non dominated criteria vectors,This enumeration problem is easy,A polynomial time algorithm for calculating a strict Pareto optimum,A polynomial number of non dominated criteria vectors.Vincent T’kindt

15 Theory of Multicriteria SchedulingThe enumeration of Pareto optima is a challenging issue,How hard is it to perform the enumeration? Complexity theory.How conflicting are the criteria? A priori evaluation, Algorithmic evaluation, A posteriori evaluation (experimental evaluation).Vincent T’kindt

17 Theory of Multicriteria SchedulingBut now what happen for multicriteria optimisation?We minimise K criteria Zi,Enumeration of strict Pareto optima,Counting problem CInput data, or instance, denoted by I (set DO).Question: how many optimal solutions are there regarding the objective of problem O?Enumeration problem EInput data, or instance, denoted by I (set DO).Goal: find the set SI the optimal solutions regarding the objective of problem O.Vincent T’kindt

18 Theory of Multicriteria SchedulingProblems which can be solved in polynomial time in the input size and number of solutionsSpatial complexityvsTemporal complexity,V. T’kindt, K. Bouibede-Hocine, C. Esswein (2007). Counting and Enumeration Complexity with application to Multicriteria Scheduling, Annals of Operations Research, 153:Vincent T’kindt

24 Theory of Multicriteria SchedulingThe maximum angle between c1 and c2 is obtained for :ui=0, i=1,…,l, and ui≥0, i=l+1,…,nandvi ≥0, i=1,…,l, and vi=0, i=l+1,…,nas the weights are non negative.This can be helpful to identify/generate instances with a potentially high number of strict Pareto optima.Vincent T’kindt

25 Theory of Multicriteria SchedulingDrawback: the number of strict Pareto optima also depends on the spreading of solutions (constraints),Drawback: not easy to generalize to max criteria.Generally, the number of strict Pareto optima is evaluated by means of an algorithmic analysis,See for instance the 1|di|Lmax, wCsum problem,But we have a bound on the number of non dominated criteria vectors.Vincent T’kindt

26 StructureTheory of Multicriteria Scheduling,Optimality definition,How to solve a multicriteria scheduling problem,Application to a bicriteria scheduling problem,Considerations about the enumeration of optimal solutions.Some models and algorithms,Scheduling with interfering job sets,Scheduling with rejection cost.Solution of bicriteria single machine problem by mathematical programmingVincent T’kindt

27 Some models and algorithmsA classification based on model features and not simply on machine configurations,Scheduling with controllable data,Scheduling with setup times,Just-in-Time scheduling,Robust and flexible scheduling,Scheduling with interfering job sets,Scheduling with rejection costs,Scheduling with completion times,Scheduling with only due date based criteria,….Vincent T’kindt

32 Scheduling with rejection costsA set of n jobs to be scheduled,A job can be scheduled or rejected,Minimize a « classic » criterion Z,Minimize the rejection cost RC=Si rci, Often Fl(Z,RC)=Z+RC is minimized.Vincent T’kindt

38 StructureTheory of Multicriteria Scheduling,Optimality definition,How to solve a multicriteria scheduling problem,Application to a bicriteria scheduling problem,Considerations about the enumeration of optimal solutions.Some models and algorithms,Scheduling with interfering job sets,Scheduling with rejection cost.Solution of bicriteria single machine problem by mathematical programmingVincent T’kindt

53 Now what’s going on?Investigation of structural properties of the Pareto set for scheduling problems,How to quickly calculate a Pareto optimum starting with a known one?Generalized dominance conditions,Measuring the conflictness of criteria: from cone dominance to the complexity of counting problems,Complexity of exponential algorithms,…Vincent T’kindt

54 Now what’s going on? Investigation of emerging models,Scheduling with interfering job sets,Scheduling with rejection costs,Scheduling for new orders,Combined models: scheduling with rejection costs and new orders, …Vincent T’kindt

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